Properties

Label 8281.2.a.u.1.1
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $1$
Dimension $2$
CM discriminant -91
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.1241179138\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 637)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 8281.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{4} -3.60555 q^{5} -3.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{4} -3.60555 q^{5} -3.00000 q^{9} +4.00000 q^{16} -3.60555 q^{19} +7.21110 q^{20} -1.00000 q^{23} +8.00000 q^{25} -5.00000 q^{29} +10.8167 q^{31} +6.00000 q^{36} -7.21110 q^{41} -9.00000 q^{43} +10.8167 q^{45} +3.60555 q^{47} +11.0000 q^{53} +14.4222 q^{59} -8.00000 q^{64} -10.8167 q^{73} +7.21110 q^{76} +15.0000 q^{79} -14.4222 q^{80} +9.00000 q^{81} +18.0278 q^{83} +3.60555 q^{89} +2.00000 q^{92} +13.0000 q^{95} +18.0278 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 6 q^{9} + O(q^{10}) \) \( 2 q - 4 q^{4} - 6 q^{9} + 8 q^{16} - 2 q^{23} + 16 q^{25} - 10 q^{29} + 12 q^{36} - 18 q^{43} + 22 q^{53} - 16 q^{64} + 30 q^{79} + 18 q^{81} + 4 q^{92} + 26 q^{95} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) −2.00000 −1.00000
\(5\) −3.60555 −1.61245 −0.806226 0.591608i \(-0.798493\pi\)
−0.806226 + 0.591608i \(0.798493\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −3.60555 −0.827170 −0.413585 0.910465i \(-0.635724\pi\)
−0.413585 + 0.910465i \(0.635724\pi\)
\(20\) 7.21110 1.61245
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 0 0
\(25\) 8.00000 1.60000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 10.8167 1.94273 0.971364 0.237595i \(-0.0763593\pi\)
0.971364 + 0.237595i \(0.0763593\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −7.21110 −1.12619 −0.563093 0.826394i \(-0.690389\pi\)
−0.563093 + 0.826394i \(0.690389\pi\)
\(42\) 0 0
\(43\) −9.00000 −1.37249 −0.686244 0.727372i \(-0.740742\pi\)
−0.686244 + 0.727372i \(0.740742\pi\)
\(44\) 0 0
\(45\) 10.8167 1.61245
\(46\) 0 0
\(47\) 3.60555 0.525924 0.262962 0.964806i \(-0.415301\pi\)
0.262962 + 0.964806i \(0.415301\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.0000 1.51097 0.755483 0.655168i \(-0.227402\pi\)
0.755483 + 0.655168i \(0.227402\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 14.4222 1.87761 0.938806 0.344447i \(-0.111934\pi\)
0.938806 + 0.344447i \(0.111934\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −10.8167 −1.26599 −0.632997 0.774154i \(-0.718175\pi\)
−0.632997 + 0.774154i \(0.718175\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 7.21110 0.827170
\(77\) 0 0
\(78\) 0 0
\(79\) 15.0000 1.68763 0.843816 0.536633i \(-0.180304\pi\)
0.843816 + 0.536633i \(0.180304\pi\)
\(80\) −14.4222 −1.61245
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 18.0278 1.97880 0.989402 0.145204i \(-0.0463840\pi\)
0.989402 + 0.145204i \(0.0463840\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.60555 0.382188 0.191094 0.981572i \(-0.438797\pi\)
0.191094 + 0.981572i \(0.438797\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.00000 0.208514
\(93\) 0 0
\(94\) 0 0
\(95\) 13.0000 1.33377
\(96\) 0 0
\(97\) 18.0278 1.83044 0.915221 0.402953i \(-0.132016\pi\)
0.915221 + 0.402953i \(0.132016\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −16.0000 −1.60000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −19.0000 −1.78737 −0.893685 0.448695i \(-0.851889\pi\)
−0.893685 + 0.448695i \(0.851889\pi\)
\(114\) 0 0
\(115\) 3.60555 0.336219
\(116\) 10.0000 0.928477
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −21.6333 −1.94273
\(125\) −10.8167 −0.967471
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −12.0000 −1.00000
\(145\) 18.0278 1.49712
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −39.0000 −3.13256
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 14.4222 1.12619
\(165\) 0 0
\(166\) 0 0
\(167\) −18.0278 −1.39503 −0.697515 0.716570i \(-0.745711\pi\)
−0.697515 + 0.716570i \(0.745711\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 10.8167 0.827170
\(172\) 18.0000 1.37249
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −25.0000 −1.86859 −0.934294 0.356504i \(-0.883969\pi\)
−0.934294 + 0.356504i \(0.883969\pi\)
\(180\) −21.6333 −1.61245
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −7.21110 −0.525924
\(189\) 0 0
\(190\) 0 0
\(191\) −20.0000 −1.44715 −0.723575 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 26.0000 1.81592
\(206\) 0 0
\(207\) 3.00000 0.208514
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) −22.0000 −1.51097
\(213\) 0 0
\(214\) 0 0
\(215\) 32.4500 2.21307
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 18.0278 1.20723 0.603614 0.797277i \(-0.293727\pi\)
0.603614 + 0.797277i \(0.293727\pi\)
\(224\) 0 0
\(225\) −24.0000 −1.60000
\(226\) 0 0
\(227\) −14.4222 −0.957235 −0.478618 0.878023i \(-0.658862\pi\)
−0.478618 + 0.878023i \(0.658862\pi\)
\(228\) 0 0
\(229\) −21.6333 −1.42957 −0.714785 0.699345i \(-0.753475\pi\)
−0.714785 + 0.699345i \(0.753475\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −29.0000 −1.89985 −0.949927 0.312473i \(-0.898843\pi\)
−0.949927 + 0.312473i \(0.898843\pi\)
\(234\) 0 0
\(235\) −13.0000 −0.848026
\(236\) −28.8444 −1.87761
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −10.8167 −0.696762 −0.348381 0.937353i \(-0.613268\pi\)
−0.348381 + 0.937353i \(0.613268\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 15.0000 0.928477
\(262\) 0 0
\(263\) −31.0000 −1.91154 −0.955771 0.294112i \(-0.904976\pi\)
−0.955771 + 0.294112i \(0.904976\pi\)
\(264\) 0 0
\(265\) −39.6611 −2.43636
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 28.8444 1.75217 0.876087 0.482154i \(-0.160145\pi\)
0.876087 + 0.482154i \(0.160145\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 17.0000 1.02143 0.510716 0.859750i \(-0.329381\pi\)
0.510716 + 0.859750i \(0.329381\pi\)
\(278\) 0 0
\(279\) −32.4500 −1.94273
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 21.6333 1.26599
\(293\) −18.0278 −1.05319 −0.526596 0.850115i \(-0.676532\pi\)
−0.526596 + 0.850115i \(0.676532\pi\)
\(294\) 0 0
\(295\) −52.0000 −3.02756
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −14.4222 −0.827170
\(305\) 0 0
\(306\) 0 0
\(307\) 32.4500 1.85202 0.926009 0.377503i \(-0.123217\pi\)
0.926009 + 0.377503i \(0.123217\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −30.0000 −1.68763
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 28.8444 1.61245
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −18.0000 −1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) −36.0555 −1.97880
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −23.0000 −1.25289 −0.626445 0.779466i \(-0.715491\pi\)
−0.626445 + 0.779466i \(0.715491\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 32.0000 1.71785 0.858925 0.512101i \(-0.171133\pi\)
0.858925 + 0.512101i \(0.171133\pi\)
\(348\) 0 0
\(349\) 32.4500 1.73701 0.868503 0.495683i \(-0.165082\pi\)
0.868503 + 0.495683i \(0.165082\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 36.0555 1.91904 0.959521 0.281638i \(-0.0908778\pi\)
0.959521 + 0.281638i \(0.0908778\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −7.21110 −0.382188
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −6.00000 −0.315789
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 39.0000 2.04135
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) −4.00000 −0.208514
\(369\) 21.6333 1.12619
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) −26.0000 −1.33377
\(381\) 0 0
\(382\) 0 0
\(383\) −28.8444 −1.47388 −0.736940 0.675958i \(-0.763730\pi\)
−0.736940 + 0.675958i \(0.763730\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 27.0000 1.37249
\(388\) −36.0555 −1.83044
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −54.0833 −2.72122
\(396\) 0 0
\(397\) 3.60555 0.180957 0.0904787 0.995898i \(-0.471160\pi\)
0.0904787 + 0.995898i \(0.471160\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 32.0000 1.60000
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −32.4500 −1.61245
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −39.6611 −1.96111 −0.980557 0.196236i \(-0.937128\pi\)
−0.980557 + 0.196236i \(0.937128\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −65.0000 −3.19072
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 0 0
\(423\) −10.8167 −0.525924
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −16.0000 −0.773389
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.60555 0.172477
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −41.0000 −1.94797 −0.973984 0.226615i \(-0.927234\pi\)
−0.973984 + 0.226615i \(0.927234\pi\)
\(444\) 0 0
\(445\) −13.0000 −0.616259
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 38.0000 1.78737
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −7.21110 −0.336219
\(461\) −7.21110 −0.335855 −0.167927 0.985799i \(-0.553707\pi\)
−0.167927 + 0.985799i \(0.553707\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) −20.0000 −0.928477
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −28.8444 −1.32347
\(476\) 0 0
\(477\) −33.0000 −1.51097
\(478\) 0 0
\(479\) 39.6611 1.81216 0.906080 0.423106i \(-0.139060\pi\)
0.906080 + 0.423106i \(0.139060\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 22.0000 1.00000
\(485\) −65.0000 −2.95150
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 40.0000 1.80517 0.902587 0.430507i \(-0.141665\pi\)
0.902587 + 0.430507i \(0.141665\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 43.2666 1.94273
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 21.6333 0.967471
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −24.0000 −1.06483
\(509\) −3.60555 −0.159813 −0.0799066 0.996802i \(-0.525462\pi\)
−0.0799066 + 0.996802i \(0.525462\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) −43.2666 −1.87761
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −28.8444 −1.24705
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 37.0000 1.58201 0.791003 0.611812i \(-0.209559\pi\)
0.791003 + 0.611812i \(0.209559\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 18.0278 0.768008
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 68.5055 2.88205
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.00000 −0.0419222 −0.0209611 0.999780i \(-0.506673\pi\)
−0.0209611 + 0.999780i \(0.506673\pi\)
\(570\) 0 0
\(571\) 3.00000 0.125546 0.0627730 0.998028i \(-0.480006\pi\)
0.0627730 + 0.998028i \(0.480006\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −8.00000 −0.333623
\(576\) 24.0000 1.00000
\(577\) 36.0555 1.50101 0.750505 0.660864i \(-0.229810\pi\)
0.750505 + 0.660864i \(0.229810\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) −36.0555 −1.49712
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18.0278 0.744085 0.372043 0.928216i \(-0.378658\pi\)
0.372043 + 0.928216i \(0.378658\pi\)
\(588\) 0 0
\(589\) −39.0000 −1.60697
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −46.8722 −1.92481 −0.962405 0.271620i \(-0.912441\pi\)
−0.962405 + 0.271620i \(0.912441\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 11.0000 0.449448 0.224724 0.974422i \(-0.427852\pi\)
0.224724 + 0.974422i \(0.427852\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 39.6611 1.61245
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 14.4222 0.579677 0.289839 0.957076i \(-0.406398\pi\)
0.289839 + 0.957076i \(0.406398\pi\)
\(620\) 78.0000 3.13256
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −43.2666 −1.71698
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 17.0000 0.671460 0.335730 0.941958i \(-0.391017\pi\)
0.335730 + 0.941958i \(0.391017\pi\)
\(642\) 0 0
\(643\) 43.2666 1.70627 0.853134 0.521691i \(-0.174699\pi\)
0.853134 + 0.521691i \(0.174699\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −34.0000 −1.33052 −0.665261 0.746611i \(-0.731680\pi\)
−0.665261 + 0.746611i \(0.731680\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −28.8444 −1.12619
\(657\) 32.4500 1.26599
\(658\) 0 0
\(659\) −19.0000 −0.740135 −0.370067 0.929005i \(-0.620665\pi\)
−0.370067 + 0.929005i \(0.620665\pi\)
\(660\) 0 0
\(661\) −10.8167 −0.420719 −0.210360 0.977624i \(-0.567463\pi\)
−0.210360 + 0.977624i \(0.567463\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 5.00000 0.193601
\(668\) 36.0555 1.39503
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −51.0000 −1.96591 −0.982953 0.183858i \(-0.941141\pi\)
−0.982953 + 0.183858i \(0.941141\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) −21.6333 −0.827170
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −36.0000 −1.37249
\(689\) 0 0
\(690\) 0 0
\(691\) 46.8722 1.78310 0.891551 0.452921i \(-0.149618\pi\)
0.891551 + 0.452921i \(0.149618\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −23.0000 −0.868698 −0.434349 0.900745i \(-0.643022\pi\)
−0.434349 + 0.900745i \(0.643022\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(710\) 0 0
\(711\) −45.0000 −1.68763
\(712\) 0 0
\(713\) −10.8167 −0.405087
\(714\) 0 0
\(715\) 0 0
\(716\) 50.0000 1.86859
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 43.2666 1.61245
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −40.0000 −1.48556
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −54.0833 −1.99761 −0.998806 0.0488615i \(-0.984441\pi\)
−0.998806 + 0.0488615i \(0.984441\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −54.0833 −1.97880
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 27.0000 0.985244 0.492622 0.870243i \(-0.336039\pi\)
0.492622 + 0.870243i \(0.336039\pi\)
\(752\) 14.4222 0.525924
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −47.0000 −1.70824 −0.854122 0.520073i \(-0.825905\pi\)
−0.854122 + 0.520073i \(0.825905\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −46.8722 −1.69911 −0.849557 0.527496i \(-0.823131\pi\)
−0.849557 + 0.527496i \(0.823131\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 40.0000 1.44715
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −32.4500 −1.17018 −0.585088 0.810970i \(-0.698940\pi\)
−0.585088 + 0.810970i \(0.698940\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 36.0555 1.29683 0.648413 0.761288i \(-0.275433\pi\)
0.648413 + 0.761288i \(0.275433\pi\)
\(774\) 0 0
\(775\) 86.5332 3.10837
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 26.0000 0.931547
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 39.6611 1.41376 0.706882 0.707331i \(-0.250101\pi\)
0.706882 + 0.707331i \(0.250101\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −10.8167 −0.382188
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −31.0000 −1.08990 −0.544951 0.838468i \(-0.683452\pi\)
−0.544951 + 0.838468i \(0.683452\pi\)
\(810\) 0 0
\(811\) 43.2666 1.51930 0.759648 0.650334i \(-0.225371\pi\)
0.759648 + 0.650334i \(0.225371\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 32.4500 1.13528
\(818\) 0 0
\(819\) 0 0
\(820\) −52.0000 −1.81592
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) −6.00000 −0.208514
\(829\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 65.0000 2.24942
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 57.6888 1.99164 0.995820 0.0913415i \(-0.0291155\pi\)
0.995820 + 0.0913415i \(0.0291155\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) 0 0
\(844\) 10.0000 0.344214
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 44.0000 1.51097
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −18.0278 −0.617259 −0.308629 0.951182i \(-0.599870\pi\)
−0.308629 + 0.951182i \(0.599870\pi\)
\(854\) 0 0
\(855\) −39.0000 −1.33377
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) −64.8999 −2.21307
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −54.0833 −1.83044
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −36.0555 −1.20723
\(893\) −13.0000 −0.435028
\(894\) 0 0
\(895\) 90.1388 3.01301
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −54.0833 −1.80378
\(900\) 48.0000 1.60000
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −53.0000 −1.75984 −0.879918 0.475125i \(-0.842403\pi\)
−0.879918 + 0.475125i \(0.842403\pi\)
\(908\) 28.8444 0.957235
\(909\) 0 0
\(910\) 0 0
\(911\) 37.0000 1.22586 0.612932 0.790135i \(-0.289990\pi\)
0.612932 + 0.790135i \(0.289990\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 43.2666 1.42957
\(917\) 0 0
\(918\) 0 0
\(919\) −20.0000 −0.659739 −0.329870 0.944027i \(-0.607005\pi\)
−0.329870 + 0.944027i \(0.607005\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −3.60555 −0.118294 −0.0591472 0.998249i \(-0.518838\pi\)
−0.0591472 + 0.998249i \(0.518838\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 58.0000 1.89985
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 26.0000 0.848026
\(941\) 61.2944 1.99814 0.999070 0.0431245i \(-0.0137312\pi\)
0.999070 + 0.0431245i \(0.0137312\pi\)
\(942\) 0 0
\(943\) 7.21110 0.234826
\(944\) 57.6888 1.87761
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 61.0000 1.97598 0.987992 0.154506i \(-0.0493785\pi\)
0.987992 + 0.154506i \(0.0493785\pi\)
\(954\) 0 0
\(955\) 72.1110 2.33346
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 86.0000 2.77419
\(962\) 0 0
\(963\) −24.0000 −0.773389
\(964\) 21.6333 0.696762
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −61.2944 −1.95499 −0.977493 0.210966i \(-0.932339\pi\)
−0.977493 + 0.210966i \(0.932339\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 9.00000 0.286183
\(990\) 0 0
\(991\) 60.0000 1.90596 0.952981 0.303029i \(-0.0979978\pi\)
0.952981 + 0.303029i \(0.0979978\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8281.2.a.u.1.1 2
7.6 odd 2 inner 8281.2.a.u.1.2 2
13.5 odd 4 637.2.c.b.246.2 yes 2
13.8 odd 4 637.2.c.b.246.1 2
13.12 even 2 inner 8281.2.a.u.1.2 2
91.5 even 12 637.2.r.b.116.1 4
91.18 odd 12 637.2.r.b.324.1 4
91.31 even 12 637.2.r.b.324.2 4
91.34 even 4 637.2.c.b.246.2 yes 2
91.44 odd 12 637.2.r.b.116.2 4
91.47 even 12 637.2.r.b.116.2 4
91.60 odd 12 637.2.r.b.324.2 4
91.73 even 12 637.2.r.b.324.1 4
91.83 even 4 637.2.c.b.246.1 2
91.86 odd 12 637.2.r.b.116.1 4
91.90 odd 2 CM 8281.2.a.u.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.c.b.246.1 2 13.8 odd 4
637.2.c.b.246.1 2 91.83 even 4
637.2.c.b.246.2 yes 2 13.5 odd 4
637.2.c.b.246.2 yes 2 91.34 even 4
637.2.r.b.116.1 4 91.5 even 12
637.2.r.b.116.1 4 91.86 odd 12
637.2.r.b.116.2 4 91.44 odd 12
637.2.r.b.116.2 4 91.47 even 12
637.2.r.b.324.1 4 91.18 odd 12
637.2.r.b.324.1 4 91.73 even 12
637.2.r.b.324.2 4 91.31 even 12
637.2.r.b.324.2 4 91.60 odd 12
8281.2.a.u.1.1 2 1.1 even 1 trivial
8281.2.a.u.1.1 2 91.90 odd 2 CM
8281.2.a.u.1.2 2 7.6 odd 2 inner
8281.2.a.u.1.2 2 13.12 even 2 inner